A converse to Littlewood-type theorem for random Dirichlet series

This article investigates random Dirichlet series within the framework of Hardy and mixed norm spaces of Dirichlet series. We completely characterize when the random symbol space of the Hardy space ${\mathcal{H}}^p$ coincides with ${\mathcal{H}}^2$. In the setting of mixed norm spaces ${\mathcal{H}}_{\alpha }^{p,q}$, the corresponding characterizations for the random symbol space of ${\mathcal{H}}_{\alpha }^{p,q}$ coinciding with ${\mathcal{H}}_{\alpha }^{2,q}$ are given when $q\ge 4$, and some sufficient and necessary conditions are established when $q<4$. We also obtain some integrability results when Dirichlet series belong to the corresponding random symbol spaces.